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Unformatted text preview: Problems in Geometry (8) 1. (A) What is the image of the line 2 x + y = 2 under Tr [- 1 , 3] ? (B) What is the image of the line x = 1 under Rot((0 , 0) ,π/ 4) ? (C) What is the image of the line x + y = 1 under Ref k , where k is the line y = 2 x ? 1 2. (A) Let P, Q be distinct points. Prove that half-turns in P and Q together give rise to a translation. Which ? (B) Consider a convex positively oriented pentagon ABCDE with positively oriented squares BAPQ, CBRS, DCTU, EDV W, AEXY constructed on its sides with respective centers M a , M b , M c , M d , M e . What is the image of A under the isometry Ψ = Rot( M d ,- π 2 ) ◦ Rot( M c ,- π 2 ) ◦ Rot( M b ,- π 2 ) ◦ Rot( M a ,- π 2 ) ? (C) What is the image of Y under Ψ ? 3. Consider a convex positively oriented quadrangle ABCD with positively oriented equilateral triangles BAP, CBQ, DCR, ADS, constructed on its sides (i. e. on the “outside”). Let α = Rot( P, π 3 ) ◦ Rot( Q, π 3 ) ◦ Rot( R, π 3 ) β = Rot( Q, π 3 )...
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- Spring '11